The double sum can be written as a Horn \(H_2\)  function [1]}
\(I=\int _{-1}^1\dfrac{(1-t)^\alpha (1+t)^\beta }{(z-t)^\lambda }P_n^{(\alpha ,\beta )}(t)dt=\dfrac{2^{\alpha +\beta +1}}{n!}\dfrac{1}{(z+1)^\lambda }\dfrac{\Gamma (\alpha +1+n)\Gamma (\beta +1+n)}{\Gamma (\alpha +\beta +n+2)} \\H_2\left(\beta +1+n,\lambda ,-n,-\beta -n,\alpha +\beta +n+2;\dfrac{2}{z+1},-1\right)\) 
